George Boole (pronounced BOOL; born November 2, 1815; died December 8, 1864) was an English self-taught mathematician, philosopher, and logician. He became the first professor of mathematics at Queen's College, Cork in Ireland. He worked in the fields of differential equations and algebraic logic and is best known for writing The Laws of Thought (1854), which includes Boolean algebra. Boolean logic, important for computer programming, helped create the foundation for the Information Age.

Boole was the son of a shoemaker. He attended primary school and learned Latin and modern languages through other means. At age 16, he began teaching to support his family. At 19, he started his own school and later ran a boarding school in Lincoln. He was active in local societies and worked with other mathematicians. In 1849, he became the first professor of mathematics at Queen's College, Cork (now University College Cork) in Ireland, where he met his future wife, Mary Everest. He continued to support social causes and stayed connected to Lincoln. In 1864, Boole died from fever-induced pleural effusion after developing pneumonia.

Boole published about 50 articles and several books during his lifetime. His key works include a paper on early invariant theory and The Mathematical Analysis of Logic, which introduced symbolic logic. He also wrote two detailed books: Treatise on Differential Equations and Treatise on the Calculus of Finite Differences. He contributed to the study of linear differential equations and the sum of residues of a rational function. In 1847, Boole created Boolean algebra, a key concept in binary logic, which became the basis for digital circuit design and modern computer science. He also worked on finding a general method for calculating probabilities based on logical connections.

Scholars such as Charles Sanders Peirce and William Stanley Jevons later expanded on Boole's work. His ideas became practical when Claude Shannon and Victor Shestakov used Boolean algebra to improve the design of electromechanical relay systems, leading to modern electronic computers. His contributions earned him honors, including the Royal Society's first gold prize for mathematics, the Keith Medal, and honorary degrees from the Universities of Dublin and Oxford. In 2015, University College Cork celebrated the 200th anniversary of Boole's birth, recognizing his lasting influence on the digital age.

Early life

George Boole was born in 1815 in Lincoln, Lincolnshire, England. He was the son of John Boole Snr (1779–1848), a shoemaker, and Mary Ann Joyce. He attended primary school and received lessons from his father. However, because his family’s business suffered, he had few more formal or academic lessons. William Brooke, a bookseller in Lincoln, may have taught him Latin, which he might also have studied at the school of Thomas Bainbridge. He taught himself modern languages. When a local newspaper published his translation of a Latin poem, a scholar accused him of copying the work, claiming he was not capable of such achievements. At age 16, Boole became the main provider for his parents and three younger siblings. He took a teaching job at Heigham’s School in Doncaster and later taught briefly in Liverpool.

Boole joined the Lincoln Mechanics’ Institute, a group founded in 1833 in Greyfriars, Lincoln. Edward Bromhead, who knew John Boole through the institute, helped George Boole with math books. Rev. George Stevens Dickson of St Swithin’s, Lincoln, gave him a calculus textbook by Sylvestre François Lacroix. Without a teacher, it took him many years to learn calculus.

At age 19, Boole started his own school in Lincoln at Free School Lane. Four years later, he took over Hall’s Academy in Waddington, outside Lincoln, after the death of Robert Hall. In 1840, he returned to Lincoln and ran a boarding school. Boole joined the Lincoln Topographical Society and served on its committee. He presented a paper titled “On the origin, progress, and tendencies of polytheism, especially amongst the ancient Egyptians and Persians, and in modern India.”

Boole became a well-known local figure and admired John Kaye, a bishop. He supported a local movement to shorten work hours. In 1847, he helped start a building society with Edmund Larken and others. He also associated with Thomas Cooper, a Chartist, whose wife was related to him.

From 1838 onward, Boole connected with British mathematicians who shared his interests. He studied algebra using symbolic methods, as they were understood at the time, and began publishing research papers.

Professorship and life in Cork

George Boole was recognized as a mathematician when he was appointed in 1849 as the first professor of mathematics at Queen's College, Cork (now University College Cork, or UCC) in Ireland. While working there, he met his future wife, Mary Everest, in 1850. She was visiting her uncle, John Ryall, who was a professor of Greek. They married in 1855. Boole continued to connect with Lincoln, where he worked with E. R. Larken on efforts to reduce prostitution.

In 1861, Boole was involved in a court case in Ireland’s Court of Queen’s Bench. The case involved John Hewitt Wheatley of Craig House, Sligo, and required him to pay £400. As a result, Wheatley’s estate and interest in lands in Maghan/Mahon, County Cork, were transferred to Boole.

In March 1863, Boole rented Litchfield Cottage in Cork, the home where he and his wife, Mary, would live until his death in December 1864. The property description in the legal documents stated, "all that and those the dwelling house called Litchfield Cottage with the premises and appurtenances thereunto belonging and the Garden and Walled in field to the rere thereof." In his will, Boole left all his "estate term and interest" in the lease of Litchfield Cottage to his wife. In August 1865, about eight months after his death, Mary (who was then living at 68 Harley Street, London) transferred the house to Francis Heard of Ballintemple, Cork, Esquire, a captain in her Majesty’s 87th Regiment of South Cork.

Honours and awards

In 1844, Boole's paper titled "On a General Method in Analysis" was awarded the first gold prize for mathematics by the Royal Society. In 1855, he was awarded the Keith Medal by the Royal Society of Edinburgh. In 1857, he was chosen as a Fellow of the Royal Society (FRS). He was also given honorary degrees (LL.D.) by the University of Dublin and the University of Oxford.

Works

George Boole's first published paper was titled "Researches in the theory of analytical transformations, with a special application to the reduction of the general equation of the second order." It appeared in the Cambridge Mathematical Journal in February 1840 (Volume 2, No. 8, pages 64–73). This work helped him become friends with Duncan Farquharson Gregory, the journal's editor. Boole wrote about 50 articles and a few other separate publications during his career.

In 1841, Boole published an important paper in early invariant theory. In 1844, the Royal Society gave him a medal for his memoir titled "On a General Method in Analysis." This work contributed to the theory of linear differential equations, moving from equations with constant coefficients to those with variable coefficients. A key idea in his methods was recognizing that operations might not always follow the same order. In 1847, Boole published The Mathematical Analysis of Logic, his first work on symbolic logic.

During his lifetime, Boole completed two major treatises on mathematics. The Treatise on Differential Equations was published in 1859, followed the next year by The Treatise on the Calculus of Finite Differences, which built on the earlier work. After Boole's death, Todhunter republished his treatises with some of Boole's revisions and added a supplement meant for a second edition.

In 1857, Boole published a treatise titled "On the Comparison of Transcendent, with Certain Applications to the Theory of Definite Integrals." In this work, he studied the sum of residues of a rational function. Among his findings was a result now called Boole's identity:

$$
sum_{k=1}^{n} frac{a_k}{a_k + b_k t} = 1
$$

for any real numbers $a_k > 0$, $b_k$, and $t > 0$. This identity has important uses in the theory of the Hilbert transform.

In 1847, Boole published a pamphlet titled Mathematical Analysis of Logic. Later, he considered this work an incomplete explanation of his logical system and preferred his later book, An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, as the final version of his ideas. Boole did not intend to challenge Aristotle's logic but aimed to organize it, give it a foundation, and expand its use. His interest in logic began during a debate about quantification, between Sir William Hamilton, who supported "quantification of the predicate," and Augustus De Morgan, who promoted a version of what is now called De Morgan duality. Boole's approach was broader than either side's and helped create what was called the "algebra of logic" tradition.

One of Boole's innovations was his principle of wholistic reference, which later logicians like Gottlob Frege and those using standard first-order logic also adopted. A 2003 article compared Aristotelian logic and Boolean logic, highlighting the importance of wholistic reference in Boole's philosophy.

Boole thought of "elective symbols" as an algebraic structure, but he did not have the modern tools of abstract algebra to describe it. His work was a starting point for the algebra of sets, a concept not familiar to him. His early efforts faced challenges, especially in defining addition.

Boole used the word "and" to represent multiplication and "or" to represent addition. However, in his original system, addition was a partial operation, meaning it only applied to certain cases, like combining separate groups in set theory. Later writers changed this interpretation, often using "or" to mean exclusive or or symmetric difference, which made addition always defined. Another possibility, which aligns with Boole's original idea, is using "or" as non-exclusive or. This ambiguity was an early challenge in the theory, reflecting modern uses of Boolean rings and Boolean algebras. Boole and Jevons debated this issue in 1863, focusing on the correct value of $x + x$. Jevons argued the result should be $x$, which is correct for disjunction. Boole left the result undefined, opposing the answer $0$, which applies to exclusive or. He believed the equation $x + x = 0$ incorrectly implied $x = 0$, a false comparison to regular algebra.

The second part of The Laws of Thought included an attempt to develop a general method for calculating probabilities. The goal was to find an algorithm that could determine the probability of any event based on the probabilities of other related events.

Death

In late November 1864, Boole walked three miles in heavy rain from his home at Lichfield Cottage in Ballintemple to the university. He lectured while still wearing his wet clothes. Soon after, he became sick with pneumonia. His wife believed that treating the cause of an illness would help, so she wrapped him in wet blankets, thinking the wet clothes had caused his sickness. His condition got worse, and on December 8, 1864, he died from a condition caused by fever. He was buried in the Church of Ireland cemetery at St Michael's, Church Road, Blackrock, which is a part of Cork. There is a plaque in the nearby church that honors him.

Legacy

George Boole is the namesake of the branch of algebra called Boolean algebra, as well as the namesake of the lunar crater Boole. The keyword "Bool" represents a Boolean data type in many programming languages, though Pascal and Java, among others, use the full name "Boolean." The library, underground lecture theatre complex, and the Boole Centre for Research in Informatics at University College Cork are named in his honor. A road called Boole Heights in Bracknell, Berkshire is named after him.

Boole's work was built upon and improved by several writers, beginning with William Stanley Jevons, who also wrote the article about Boole in the Encyclopædia Britannica. Augustus De Morgan studied the logic of relations, and Charles Sanders Peirce combined his work with Boole's during the 1870s. Other important figures included Platon Sergeevich Poretskii and William Ernest Johnson. The idea of applying Boolean algebra to statements in propositional calculus is credited to Hugh MacColl in 1877, a concept later reviewed by Johnson. Surveys of these developments were published by Ernst Schröder, Louis Couturat, and Clarence Irving Lewis.

In 1921, economist John Maynard Keynes wrote a book on probability theory called A Treatise of Probability. Keynes believed Boole made a mistake in defining independence, which weakened much of his analysis. Later, David Miller wrote a book called The Last Challenge Problem, which provided a method that aligns with Boole's system and attempted to solve problems identified by Keynes and others. Earlier, Theodore Hailperin showed that Boole had used the correct mathematical definition of independence in his examples.

At first, Boole's work and that of later logicians seemed to have no practical use. Claude Shannon studied philosophy at the University of Michigan, where he learned about Boole's ideas. Shannon realized Boole's work could be used to create real-world mechanisms and processes, making it highly useful. In 1937, Shannon wrote a master's thesis at the Massachusetts Institute of Technology, showing how Boolean algebra could improve the design of electromechanical relays used in telephone switches. He also proved that relay circuits could solve Boolean algebra problems. Using electrical switches to process logic is the basic idea behind modern electronic computers. Earlier, in 1935, Victor Shestakov at Moscow State University proposed a theory of electric switches based on Boolean logic. However, his findings were first published in 1941. Because of these developments, Boolean algebra became the foundation of practical digital circuit design, and Boole's ideas, through Shannon and Shestakov, became the basis for the Information Age.

George Boole's influence is everywhere today, in computers, information storage, electronic circuits, and systems that support life, learning, and communication in the 21st century. His important contributions to mathematics, logic, and probability laid the essential groundwork for modern mathematics, microelectronic engineering, and computer science.

In 2015, the 200th anniversary of Boole's birth was celebrated. University College Cork joined people worldwide to honor his life and achievements. The university's "George Boole 200" project included events, student activities, and academic conferences about Boole's legacy in the digital age. A new edition of Desmond MacHale's 1985 biography, The Life and Work of George Boole: A Prelude to the Digital Age, was published in 2014.

Google marked Boole's 200th birthday on November 2, 2015, with a special algebraic version of its Google Doodle. In September 2022, a statue of George Boole, showing him as a teacher, was unveiled at Lincoln Central Train Station in Lincoln, his hometown.

Views

George Boole shared his ideas in four speeches: The Genius of Sir Isaac Newton, The Right Use of Leisure, The Claims of Science, and The Social Aspect of Intellectual Culture. The first speech was given in 1835 after Charles Anderson-Pelham, 1st Earl of Yarborough, donated a statue of Isaac Newton to the Mechanics' Institute in Lincoln. The second speech, delivered in 1847, praised the success of a movement in Lincoln to shorten work hours, led by Alexander Leslie-Melville of Branston Hall. The Claims of Science was presented in 1851 at Queen's College, Cork. The Social Aspect of Intellectual Culture was given in Cork in 1855 to the Cuvierian Society.

Although his biographer, Des MacHale, described Boole as an "agnostic deist," Boole studied many Christian religious writings. He connected his interest in mathematics and theology by comparing the Christian idea of the Trinity (Father, Son, and Holy Spirit) to the three dimensions of space. He was interested in the Jewish belief that God is a single, absolute being. Boole considered converting to Judaism but ultimately chose Unitarianism. He opposed what he called "prideful" skepticism and supported the belief in a "Supreme Intelligent Cause." He stated, "I firmly believe, for the accomplishment of a purpose of the Divine Mind." He also wrote, "To infer the existence of an intelligent cause from the evidence of purpose in the world around us… these are the most solid foundations, apart from religious revelation, of the belief that the world is not left to chance and fate."

Boole's wife, Mary Everest Boole, later said two influences shaped his work: a universal mysticism influenced by Jewish thought and Indian logic. She noted that a mystical experience during his youth helped guide his life's work. In Laws of Thought, Chapter 13, Boole used examples from Baruch Spinoza and Samuel Clarke. The book briefly discusses the relationship between logic and religion, but these ideas are not fully explained. Boole seemed surprised by how people received his work as a mathematical tool.

Mary Boole also claimed that Indian thought, especially Indian logic, had a deep influence on George Boole, as well as on Augustus De Morgan and Charles Babbage, through her uncle, George Everest.

Family

In 1855, Boole married Mary Everest, who was the niece of George Everest. Mary later wrote several educational works about her husband's ideas.

The Booles had five daughters:

• Mary Ellen (1856–1908) married Charles Howard Hinton, a mathematician and author. They had four children. After Charles died suddenly in April 1907, Mary Ellen died by suicide in Washington, D.C., in May 1908. Their children included George Hinton (1882–1943), a mining engineer and botanist; H. E. Hinton (1912–1977), an entomologist; Geoffrey Hinton (born 1947), a cognitive psychologist and computer scientist who won the Nobel Prize for physics in 2024 for work on artificial neural networks; Eric Hinton (born 1884); William Hinton (1886–1909); and Sebastian Hinton (1887–1923), a lawyer who invented the jungle gym. Other family members included Jean Hinton (married name Rosner) (1917–2002), a peace activist, and William H. Hinton (1919–2004), who visited China in the 1930s and 1940s and wrote an influential account of Communist land reform. Joan Hinton (1921–2010) worked on the Manhattan Project and lived in China from 1948 until her death on June 8, 2010. She was married to Sid Engst.

• Margaret (1858–1935) married Edward Ingram Taylor, an artist. Their son Geoffrey Ingram Taylor became a mathematician and a Fellow of the Royal Society. Their other son, Julian Taylor, was a professor of surgery.

• Alicia (1860–1940) made important contributions to four-dimensional geometry. Her son Leonard Stott, a medical doctor and tuberculosis pioneer, invented a portable X-ray machine, a pneumothorax apparatus, and a system of navigation based on spherical coordinates.

• Lucy Everest (1862–1904) was the first female professor of chemistry in England.

• Ethel Lilian (1864–1960) married Wilfrid Michael Voynich, a Polish scientist and revolutionary. She was the author of the novel The Gadfly.